Synchronous or Clear-Path Control
in Personal Rapid Transit Systems
J. Edward Anderson
An equation is derived for the ratio of the expected value of the maximum possible station flow to average line flow in a personal rapid transit or dual-mode system using fully synchronous control. It is shown that such a system is impractical except in very small networks.
In a synchronously controlled PRT system a vehicle waits at an origin station until the path through all merges to the destination is clear. The system then reserves this path for the specific vehicle in question, which then proceeds without any maneuvers directly to the destination. Such a scheme was proposed in the late 1960s for the automated transit system deployed in Morgantown, West Virginia. Because of its inflexibility in face of failure of any vehicle to maintain its path, interest declined, but the clear-path idea has emerged again on the Transit Alternatives mailing list on Internet. It is thus worthwhile to show why, by estimating the wait time for an origin-to-destination reservation, fully synchronous systems are not practical except in very small networks.
Let f be the average rush-period line flow in vehicles per second in a PRT system, let h be the minimum time headway in seconds, and let m be the number of merge points to be passed on an average trip including the merge from station to line. Then, 1/h is the number of moving slots passing any point on the main line per second, and the quantityfh is the average fraction of line slots occupied by vehicles. Thus the probability of finding an empty slot at any instant of time is 1 - fh.
To find a clear path, it is necessary to find m slots at the time a vehicle is ready to leave a station. The probability of such an event is the probability of the simultaneous occurrence of m independent events, which is the product of the probabilities of the individual events. Thus, the probability of finding a clear path at any time is
(1)
If the search is repeated n times, the probability of finding a clear path is n times the probabilities of the individual events. If there is to be a 50-50 chance of attaining a clear path in n tries, its probability of occurrence must be one half. Thus, set
or
(2)
If at any moment a clear path is not attained, it is necessary to wait h seconds to try again. Thus, the expected wait time is nh or
(3)
In dimensionless terms, the quantity is the ratio of the maximum station flow, restricted by the need to wait for a clear path, to the average line flow. Thus
. (4)
Equation (4) is plotted in Figure 1 with fh, the average line-slot occupancy ratio, as the abscissa, for a range of values of m. For economic viability in a PRT system, it is necessary to be able to operate at values of the average line-slot occupancy ratio up to about one half. To obtain sufficient throughput at the largest stations, it is necessary that the ratio of maximum station flow to maximum line flow be at least a quarter. From Figure 1, it is seen therefore that if clear-path control is to be practical, there should be no more than an average of four merges, including station-to-line merges.
The idea, sometimes advanced, of a large PRT network using clear-path or synchronous control is seen to be completely impractical.
Figure 1. Performance of a PRT System
Using Synchronous Control.